546 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
in terms of the other invariants), the coefficients (X, /a, gJ, X) are expressed in terms 
of g, h, k, that is of A, B, C, viz. we have 
[ 72^/Fx =h{g -16£) 2 -9%+16£)+(^-16%/A, 
24 v /&> =9F+1 6hk -gh-^/A, 
J 24 \/k 3 gJ=9k 2 +1 61 iJc -gh-\-\/ A, 
I 72 N /£y=% -16kf-^k{g-\-16k)-{g-\6k)^A-, 
these values of ( X , gj, gJ, X 1 ) could of course be at once expressed in terms of (J, K, L), 
but I have not thought it necessary to make the transformation. 
317. It has been already noticed that the linear covariant (No. 15, = P#-f-Qy), was 
=\/A (Sic, yixx, Y), 
it is to be added that the septic covariant (PTr + Q ! y) is 
-nAXX, Y), 
and that the canonical forms of the cubicovariants <p,(#, y ), &c. are as follows: 
<P,(X, Y) =- s /A (p., Wh 3 Sic, p’JX, Y) 3 , 
<P,(X, Y) = A (fc, Jk, - SX -f^'IX, Y) 3 , 
Y)} - Sk, - Sh /*' IX, Y)>, 
{ *p.(X, Y) } = Aty, - 3^/T, 3 -pc’ XX, Y)», 
^,(X, Y)=yA 5 |' (2^7-3 fJc +(«.y),T 
| 3( S k'-\-Ss/k—‘lpk)’ L 
j — 3( si?+w'Sii- v*). r ’ ’ ’ 
(- vl/F-Wi + w »“), j 
4>,(X, Y)=S&(^, -Sk, 5f*XX, Y) 3 , 
<P.(X, Y) =N /A 3 ' 
' 
(7,/Afn + 96(2^/7-3 fJc + 
-3(3 v /A N /r-96( )). 
+ 3(3\/ A S k — 96( 7 -f- ijm! k — ‘ZuJ k , 
- (7v/V +96(2^/77-377: + W -7|) 
V 
(X, Y)» 
J 
or, as the last formula may also be written, 
<p 4 (X, Y)= x /A 5 j' ((7^~ 53A+110£> -64tyV*)> 
j— 3((3^+151A— 90k)*/k-64x ! g, 2 ), 
| +3((3^+151A- 90£\/&-64x f A' 2 ), 
[- ((7^- 53/4+110%' -64x>>/£) 
) X, Y) 3 . 
j 
